# A Low-Rank Tensor Factorization Using Implicit Similarity in Trust Relationships

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Related Studies

#### 2.1. User-Based Collaborative Filtering

#### 2.2. Low-Rank Tensor Factorization

#### 2.3. Algorithm Optimization

## 3. Method

#### 3.1. Model Architecture

#### 3.2. User-Based Collaborative Filtering

#### 3.3. Tensor Tucker Factorization Model

#### 3.4. A Low-Rank Tensor Factorization Using Implicit Similarity in Trust Relationships (LTF-ISTR)

- Step 1: Select the initial point ${X}_{0}$, given the precision $eps$, and let $k=0$;
- Step 2: Calculate ${g}_{0}=\nabla f\left({X}_{0}\right)$, if ${g}_{0}=0$, stop; otherwise, ${d}_{0}=-{g}_{0}$;
- Step 3: Calculate ${\lambda}_{k}=-\frac{{g}_{k}^{T}{d}_{k}}{{d}_{k}^{T}A{d}_{k}}$;
- Step 4: Calculate ${X}_{k+1}={X}_{k}+{\lambda}_{k}{d}_{k}$;
- Step 5: Calculate ${g}_{k+1}=\nabla f\left({X}_{k+1}\right)$, if ${g}_{k+1}=0$, stop;
- Step 6: Calculate ${d}_{k+1}=-{g}_{k+1}+\frac{{g}_{k+1}^{T}A{d}_{k+1}}{{d}_{k}^{T}A{d}_{k}}{d}_{k}$;
- Step 7: Let $k=k+1$ and go to Step 3.

- Step 1: Select the initial point ${u}_{im}$, given error accuracy $\mathcal{E}>0$, let $k=1$;
- Step 2: Calculate $\frac{\partial L}{\partial {u}_{im}}=\nabla f\left({u}_{im}\right)$, if $\parallel \frac{\partial L}{\partial {u}_{im}}\parallel \text{}\text{}E$, stop. Otherwise, take the next step;
- Step 3: Construct search direction, let ${d}^{im}\text{}=\text{}-\frac{\partial L}{\partial {u}_{im}}\text{}+\text{}\beta {d}^{im}$ and $\beta =\frac{{\left(\frac{\partial L}{\partial {u}_{\left(i+1\right)\left(m+1\right)}}\right)}^{T}\frac{\partial L}{\partial {u}_{\left(i+1\right)\left(m+1\right)}}}{{\left(\frac{\partial L}{\partial {u}_{im}}\right)}^{T}\frac{\partial L}{\partial {u}_{im}}}$;
- Step 4: let ${u}_{\left(i+1\right)\left(m+1\right)}={u}_{im}+{\lambda}_{k}{d}^{im}$ and the Wolfe linear search criterion is used to calculate the step size ${\lambda}_{k}$. A new iteration point is obtained. Return to Step 2.$${u}_{im}\leftarrow \{\begin{array}{c}{u}_{im}+{\lambda}_{k}{d}^{im}\\ {d}^{im}=-\frac{\partial L}{\partial {u}_{im}}+\beta {d}^{im}\\ \beta =\frac{{\left(\frac{\partial L}{\partial {u}_{\left(i+1\right)\left(m+1\right)}}\right)}^{T}\frac{\partial L}{\partial {u}_{\left(i+1\right)\left(m+1\right)}}}{{\left(\frac{\partial L}{\partial {u}_{im}}\right)}^{T}\frac{\partial L}{\partial {u}_{im}}}\end{array}$$$${u}_{jn}\leftarrow \{\begin{array}{c}{u}_{jn}+{\lambda}_{k}{d}^{jn}\\ {d}^{jn}=-\frac{\partial L}{\partial {u}_{jn}}+\beta {d}^{jn}\\ \beta =\frac{{\left(\frac{\partial L}{\partial {u}_{\left(j+1\right)\left(n+1\right)}}\right)}^{T}\frac{\partial L}{\partial {u}_{\left(j+1\right)\left(n+1\right)}}}{{\left(\frac{\partial L}{\partial {u}_{jn}}\right)}^{T}\frac{\partial L}{\partial {u}_{jn}}}\end{array}$$$${c}_{kl}\leftarrow \{\begin{array}{c}{c}_{kl}+{\lambda}_{k}{d}^{kl}\\ {d}^{kl}=-\frac{\partial L}{\partial {c}_{kl}}+\beta {d}^{kl}\\ \beta =\frac{{\left(\frac{\partial L}{\partial {c}_{\left(k+1\right)\left(l+1\right)}}\right)}^{T}\frac{\partial L}{\partial {c}_{\left(k+1\right)\left(l+1\right)}}}{{\left(\frac{\partial L}{\partial {c}_{kl}}\right)}^{T}\frac{\partial L}{\partial {c}_{kl}}}\end{array}$$$${{A}^{\prime}}_{mnl}\leftarrow \{\begin{array}{c}{{A}^{\prime}}_{mnl}+{\lambda}_{k}{d}^{mnl}\\ {d}^{mnl}=-\frac{\partial L}{\partial {{A}^{\prime}}_{mnl}}+\beta {d}^{mnl}\\ \beta =\frac{{\left(\frac{\partial L}{\partial {{A}^{\prime}}_{\left(m+1\right)\left(n+1\right)\left(l+1\right)}}\right)}^{T}\frac{\partial L}{\partial {{A}^{\prime}}_{\left(m+1\right)\left(n+1\right)\left(l+1\right)}}}{{\left(\frac{\partial L}{\partial {{A}^{\prime}}_{mnl}}\right)}^{T}\frac{\partial L}{\partial {{A}^{\prime}}_{mnl}}}\end{array}$$

## 4. Experiments

#### 4.1. Dataset

#### 4.2. Evaluation Index

#### 4.3. Experimental Results and Analysis

#### 4.3.1. Experiment 1: Parameter Value

#### 4.3.2. Experiment 2: Comparison between the Fletcher-Reeves CG Method and SGD Method

#### 4.3.3. Experiment 3: Algorithm Comparison

#### 4.3.4. Experiment 4: Comparison of the Sparseness of Different Ratings

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Table 1.**Symbol description in Figure 1.

Symbol | Description |
---|---|

T | Third-order tensor |

${\mathit{U}}_{\mathit{i}}$,${\mathit{U}}_{\mathit{j}}$ | User set |

${\mathit{C}}_{\mathit{k}}$ | User implicit similarity |

A | Approximate tensor |

A′ | Low-rank tensor with implicit similarity |

${\mathit{U}}_{{\mathit{I}}^{\prime}}$,${\mathit{U}}_{{\mathit{J}}^{\prime}}$ | User factor matrix |

I’, J’ | Set of users with implicit similarity |

W_{AB} | W_{AC} | W_{AD} | W_{BC} | W_{BD} | W_{CD} |
---|---|---|---|---|---|

1.356 | 0.698 | 0 | 1.356 | 1.661 | 1.356 |

Dataset | Ciao | Filmtrust |
---|---|---|

Users | 7375 | 1508 |

Items | 99,746 | 2071 |

Users-Items Ratings | 278,483 | 35,497 |

Ratings Scale | 1–5 | 0.5–4 |

Density | 0.038% | 1.14% |

Users Trust | 111,781 | 1853 |

Learning Rate α | 0.1 | 0.01 | 0.001 | 0.0001 |

MAE | Nor | Nor | 0.688 | 1.124 |

RMSE | Nor | Nor | 0.911 | 1.501 |

Learning Rate α | 0.1 | 0.01 | 0.001 | 0.0001 |

MAE | Nor | Nor | 0.721 | 1.242 |

RMSE | Nor | Nor | 0.933 | 1.574 |

ITERATIONS | 20 | 40 | 60 | 80 | 100 |

MAE | 0.688 | 0.676 | 0.640 | 0.621 | 0.604 |

RMSE | 0.911 | 0.881 | 0.838 | 0.820 | 0.804 |

Time~(s) | 748.4 | 1372.5 | 2007.3 | 2894.6 | 3353.7 |

**Table 7.**Experimental results and computation time of MAE and RMSE at different iterations on FilmTrust.

ITERATIONS | 20 | 40 | 60 | 80 | 100 |

MAE | 0.721 | 0.644 | 0.615 | 0.601 | 0.589 |

RMSE | 0.932 | 0.834 | 0.796 | 0.778 | 0.765 |

Time~(s) | 625.8 | 1192.4 | 1810.7 | 2397.8 | 2942.0 |

MAE | RMSE | |
---|---|---|

LTF-ISTR (SGD) | 0.651 | 0.845 |

LTF-ISTR (Fletcher-Reeves CG) | 0.640 | 0.838 |

MAE | RMSE | |
---|---|---|

LTF-ISTR (SGD) | 0.627 | 0.799 |

LTF-ISTR (Fletcher-Reeves CG) | 0.615 | 0.796 |

Method | Description |
---|---|

Tensor factorization (TF) | Score prediction based on tensor factorization [36]. |

Singular value Decomposition (SVD) | Collaborative filtering algorithm based on singular value decomposition [51]. |

Singular value Decomposition plus plus (SVDpp) | A singular value decomposition algorithm that incorporates the user’s implicit behavior on items [52]. |

Baseline only | Standard line estimates that predict a given user and project [53]. |

K-nearest neighbor baseline (KNNBaseline) | Considering the deviation of the user’s score, the deviation is calculated based on the baseline [54]. |

Nonnegative matrix factorization (NMF) | Collaborative filtering algorithm based on nonnegative matrix factorization [55]. |

MAE | RMSE | Time (s) | |
---|---|---|---|

LTF-ISTR | 0.640 | 0.838 | 2077.3 |

TF | 0.697 | 0.898 | 4236.6 |

SVDpp | 0.729 | 0.941 | 54.9 |

SVD | 0.735 | 0.955 | 3.9 |

BaselineOnly | 0.737 | 0.949 | 0.7 |

KNNBaseline | 0.746 | 0.995 | 5.9 |

NMF | 0.866 | 1.121 | 6.7 |

MAE | RMSE | Time (s) | |
---|---|---|---|

LTF-ISTR | 0.615 | 0.796 | 1810.7 |

SVDpp | 0.631 | 0.813 | 8.0 |

SVD | 0.638 | 0.818 | 1.4 |

BaselineOnly | 0.644 | 0.817 | 0.5 |

KNNBaseline | 0.640 | 0.838 | 5.5 |

NMF | 0.677 | 0.879 | 1.7 |

**Table 13.**Results of MAE and RMSE for each algorithm with the Ciao dataset density of 3%, 2%, and 1%.

MAE | RMSE | ||
---|---|---|---|

density = 3% | LTF-ISTR | 0.624 | 0.836 |

TF | 0.697 | 0.898 | |

SVDpp | 0.729 | 0.941 | |

SVD | 0.735 | 0.955 | |

BaselineOnly | 0.737 | 0.949 | |

KNNBaseline | 0.746 | 0.995 | |

NMF | 0.866 | 1.211 | |

density = 2% | LTF-ISTR | 0.647 | 0.852 |

TF | 0.615 | 0.946 | |

SVDpp | 0.789 | 1.009 | |

SVD | 0.808 | 1.014 | |

BaselineOnly | 0.784 | 0.995 | |

KNNBaseline | 0.834 | 1.063 | |

NMF | 0.881 | 1.135 | |

density = 1% | LTF-ISTR | 0.695 | 0.973 |

TF | 0.638 | 1.045 | |

SVDpp | 0.796 | 1.033 | |

SVD | 0.813 | 1.028 | |

BaselineOnly | 0.810 | 1.031 | |

KNNBaseline | 0.871 | 1.154 | |

NMF | 0.959 | 1.276 |

**Table 14.**Results of MAE and RMSE for each algorithm with a FilmTrust dataset density of 3%, 2%, and 1%.

MAE | RMSE | ||
---|---|---|---|

density = 3% | LTF-ISTR | 0.575 | 0.754 |

SVDpp | 0.574 | 0.747 | |

SVD | 0.570 | 0.729 | |

BaselineOnly | 0.643 | 0.823 | |

KNNBaseline | 0.577 | 0.762 | |

NMF | 0.587 | 0.765 | |

density = 2% | LTF-ISTR | 0.614 | 0.810 |

SVDpp | 0.636 | 0.804 | |

SVD | 0.634 | 0.821 | |

BaselineOnly | 0.696 | 0.803 | |

KNNBaseline | 0.653 | 0.826 | |

NMF | 0.645 | 0.849 | |

density = 1% | LTF-ISTR | 0.637 | 0.835 |

SVDpp | 0.641 | 0.840 | |

SVD | 0.742 | 0.840 | |

BaselineOnly | 0.670 | 0.893 | |

KNNBaseline | 0.645 | 0.879 | |

NMF | 0.649 | 0.893 |

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Ma, P.; Wang, L.; Qin, J.
A Low-Rank Tensor Factorization Using Implicit Similarity in Trust Relationships. *Symmetry* **2020**, *12*, 439.
https://doi.org/10.3390/sym12030439

**AMA Style**

Ma P, Wang L, Qin J.
A Low-Rank Tensor Factorization Using Implicit Similarity in Trust Relationships. *Symmetry*. 2020; 12(3):439.
https://doi.org/10.3390/sym12030439

**Chicago/Turabian Style**

Ma, Pei, Liejun Wang, and Jiwei Qin.
2020. "A Low-Rank Tensor Factorization Using Implicit Similarity in Trust Relationships" *Symmetry* 12, no. 3: 439.
https://doi.org/10.3390/sym12030439